Calculation of a famous arithmetic series (Part 3)

In this post, we’ll consider the calculation of a very famous arithmetic series… not because the series is particularly important, but because it’s part of a legendary story about one of the greatest mathematicians who ever lived. My frank opinion is that every math teacher should know this story. While I’m not 100% certain about small details of the story — like whether young Gauss was 9 or 10 years old when the following event happened — I’m just going to go with the story as told by the website http://www.math.wichita.edu/history/men/gauss.html.

Carl Friedrich Gauss (1777-1855) is considered to be the greatest German mathematician of the nineteenth century. His discoveries and writings influenced and left a lasting mark in the areas of number theory, astronomy, geodesy, and physics, particularly the study of electromagnetism.

Gauss was born in Brunswick, Germany, on April 30, 1777, to poor, working-class parents. His father labored as a gardner and brick-layer and was regarded as an upright, honest man. However, he was a harsh parent who discouraged his young son from attending school, with expectations that he would follow one of the family trades. Luckily, Gauss’ mother and uncle, Friedrich, recognized Carl’s genius early on and knew that he must develop this gifted intelligence with education.

While in arithmetic class, at the age of ten, Gauss exhibited his skills as a math prodigy when the stern schoolmaster gave the following assignment: “Write down all the whole numbers from $1$ to $100$ and add up their sum.” When each student finished, he was to bring his slate forward and place it on the schoolmaster’s desk, one on top of the other. The teacher expected the beginner’s class to take a good while to finish this exercise. But in a few seconds, to his teacher’s surprise, Carl proceeded to the front of the room and placed his slate on the desk. Much later the other students handed in their slates.

At the end of the classtime, the results were examined, with most of them wrong. But when the schoolmaster looked at Carl’s slate, he was astounded to see only one number: $5050$ . Carl then had to explain to his teacher that he found the result because he could see that, $1+100=101$ , $2+99=101$ , $3+98=101$ , so that he could find $50$ pairs of numbers that each add up to $101$ . Thus, $50$ times $101$ will equal $5050$ .